From cube-lovers-errors@mc.lcs.mit.edu Wed Sep 3 18:42:04 1997
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Mail-from: From jbryan@pstcc.cc.tn.us Wed Sep 3 13:55:04 1997
Date: Wed, 03 Sep 1997 13:51:02 -0400 (Eastern Daylight Time)
From: Jerry Bryan
Subject: Re: Open and Closed Subgroups of G
In-Reply-To: <970901163210.20217b13@iccgcc.cle.ab.com>
To: SCHMIDTG@iccgcc.cle.ab.com
Cc: cube-lovers@ai.mit.edu
Message-Id:
On Mon, 1 Sep 1997 SCHMIDTG@iccgcc.cle.ab.com wrote:
> It might also be helpful for someone to cover the basics of cube
> parity. Although I think I understand the basic group theoretic
> concepts of permutation parity, the asymmetry of the marked faces
> of the cube have never quite left me feeling comfortable about
> how this concept is applied to the cube. Hofstadter, covers this,
> but does not discuss it in enough detail for one to fully grasp
> the concept.
>
I'll take your question as literal, assuming you mean just parity and
not twist and flip, and assuming you know the basic group theoretic
concepts of permutation parity.
Parity of the cube is best described (I think) as applying to whole
cubies rather than to facelets. As such, a quarter turn of any face is
a 4-cycle on the corner cubies and a 4-cycle on the edge cubies. A
4-cycle is odd, which is to say that it can be decomposed into an odd
number of 2-cycles. The "obvious" way to decompose a 4-cycle is into
three 2-cycles. Although decomposition of a 4-cycle into 2-cycles is
not unique, any such decomposition will contain an odd number of
2-cycles.
Start is even for both the edges and the corners (the identity consists
of zero 2-cycles). If you any quarter turn from Start, both edges and
corners become odd. Make another quarter turn, both edges and corners
become even. Make another quarter turn, both edges and corners become
odd. Etc. Edges and corners are either both even or both odd.
In the constructable group, you can have odd corners with even edges or
vice versa. For example, remove two edge cubies from a cube and
exchange them without moving any of the other cubies around. You will
be changing the parity of the edges without changing the parity of the
corners.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us
Pellissippi State (423) 539-7198
10915 Hardin Valley Road (423) 694-6435 (fax)
P.O. Box 22990
Knoxville, TN 37933-0990